First thing I would double check is if you have updated the secure_base_url configuration in the admin of Magento.
If you can provide additional details on ‘stops working’ that would help us to narrow down the problem. Do you get any error message, or does the https just turn to http?
Sounds like the SSL is installed correctly, since it works on the main page - but you could have mixed content (ie image not being called via https - *hardcoded url*) which would provide an error message, or it is as simple as you have not configured the application for using SSL by updating the secure_base_url in the Magento admin configuration.
If you want to provide the domain or send me a PM I’ll take a closer look at it.